MathOverflow is a question and answer site for professional mathematicians. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

Shimura (Crelle 221, 1966) considers the elliptic curve $E:y^2+y=x^3-x^2$ (although he doesn't use this equation) of conductor $11$ whose associated modular form is $$ q\prod_{k=1}^{+\infty}(1-q^k)^2(1-q^{11k})^2=\sum_{n=1}^{+\infty}c_nq^n $$ where $q=e^{2i\pi\tau}$ and $\tau$ is in the upper half of $\bf C$. For a prime $l$, he denotes by $K_l$ the extension of $\bf Q$ obtained by adjoining the $l$-torsion points of $E$ and shows that if $l\in[7,97]$, then ${\rm Gal}(K_l|{\bf Q})$ is isomorphic to ${\rm GL}_2({\bf F}_l)$.

Question. Is ${\rm Gal}(K_l|{\bf Q})$ now known to be isomorphic to ${\rm GL}_2({\bf F}_l)$ even for $l>97$ ?

Even if the faithful representation ${\rm Gal}(K_l|{\bf Q})\rightarrow{\rm GL}_2({\bf F}_l)$ fails to be surjective for a few $l>97$, does the recent proof of Serre's modularity conjecture not imply the

Statement. For every prime $l>5$ and every prime $p\neq11,l$, the characteristic polynomial of ${\rm Frob}_p$ (thought of as an element of ${\rm GL}_2({\bf F}_l)$) is $\equiv X^2-c_pX+p \pmod l$ ?

Shimura shows this only for $l\in[7,97]$.

Addendum. (2010/07/24) Looking at Shimura's paper beyond the first page shows that he actually proves (Section 3) that the characteristic polynomial for the action of ${\rm Frob}_p$ on the $l$-adic Tate module $T_l(E)$ is

$X^2-a_pX+p\in{\bf Z}_l[X]$

for all primes $l$ and $p\neq11, l$ and (Section 6) that $a_p=c_p$ for all $p\neq11$. And yes, he does use the Eichler-Shimura relation.

share|cite|improve this question
up vote 5 down vote accepted

The first question is answered in Serre's [Propriétés galoisiennes des points d'ordre fini des courbes elliptiques, Invent. Math. 15:4 (1972) 259--331], on page 304, section 5.2, exactly for this curve. In general this paper give a good way to determine for which $\ell$ the mod-$\ell$ representation is not surjective. Sage can do that efficiently for a given curve.

For every prime $p$ different from $\ell$ and $11$, the characteristic polynomial of $\rm{Frob}_p$ is indeed $T^2 - c_p T +p$ in $ \mathbb F_{\ell}[T]$. The isomorphism $\rm{Gal}(K _{\ell}/\mathbb Q) \to \rm{Aut}(E[\ell])=\rm{GL} _2(\mathbb F _{\ell})$ sends $\rm{Frob}_p$ to the Frobenius endomorphism $\phi:E[\ell] \to E[\ell]$ on $E/\mathbb{F}_p$. Your $c_p$ is the trace of $ \phi$ and $p$ is the determinant of it since the Eichler--Shimura relation shows that $c_p$ is the Fourier coefficent of the associated modular form. See this answer for why it is so.

share|cite|improve this answer
Many thanks for for the reference to Serre: I should have looked it up before asking the question. What I found strange was that Shimura was claiming the "Statement" only for $l\in[7,97]$, whereas it seems to follows for all $l$ from the fact that the representation $\rho_l:{\rm Gal}(K_l|{\bf Q})\to{\rm GL}_2({\bf Z}_l)$ is unramfied at every $p\neq11,l$, that for these $p$ the characteristic polynomial of ${\rm Frob}_p\in{\rm GL}_2({\bf Z}_l)$ is $T^2-a_pT+p$ where $a_p$ is defined by ${\rm Card}(E({\bf F}_l))=1-a_p+p$, and finally the fact that $a_p=c_p$. – Chandan Singh Dalawat Jul 20 '10 at 14:27
His method seems to need the restriction $l\in[7,97]$ for the surjectivity ${\rm Gal}(K_l|{\bf Q})\to{\rm GL}_2({\bf F}_l)$, though. – Chandan Singh Dalawat Jul 20 '10 at 14:27
I think the best reference is Cor.1 on p.308 of Serre's paper you mention. It says that if $E$ is a semistable elliptic curve over ${\bf Q}$ and if $p$ is the smallest prime where $E$ has good reduction, then ${\rm Gal}(K_l|{\bf Q})\to{\rm GL}_2({\bf F}_l)$ is surjective for every prime $l>(\sqrt p+1)^2$. – Chandan Singh Dalawat Jul 20 '10 at 14:49

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.